A Brief Table of Integrals

Basic Forms

1. $$\int kdx=kx+C$$

2. $$\int x^ndx=\frac{x^{n+1}}{n+1}+C,n\neq -1$$

3. $$\int\frac{dx}{x}=\ln|x|+C$$

4. $$\int e^xdx=e^x+C$$

5. $$\int a^xdx=\frac{a^x}{\ln a}+C$$

6. $$\int\sin xdx=-\cos x+C$$

7. $$\int\cos xdx=\sin x+C$$

8. $$\int\sec^2xdx=\tan x+C$$

9. $$\int\csc^2xdx=-\cot x+C$$

10. $$\int\sec x\tan xdx=\sec x+C$$

11. $$\int\csc x\cot xdx=-\csc x+C$$

12. $$\int\tan xdx=\ln|\sec x|+C$$

13. $$\int\cot xdx=\ln|\sin x|+C$$

14. $$\int\sinh xdx=\cosh x+C$$

15. $$\int\cosh xdx=\sinh x+C$$

16. $$\int\frac{dx}{\sqrt{a^2-x^2}}=\sin^{-1}\frac{x}{a}+C$$

17. $$\int\frac{dx}{a^2+x^2}=\frac{1}{a}\tan^{-1}\frac{x}{a}+C$$

18. $$\int\frac{dx}{x\sqrt{x^2-a^2}}=\frac{1}{a}\sec^{-1}\bigg|\frac{x}{a}\bigg|+C$$

19. $$\int\frac{dx}{\sqrt{a^2+x^2}}=\sinh^{-1}\frac{x}{a}+C(a>0)$$

20. $$\int\frac{dx}{\sqrt{x^2-a^2}}=\cosh^{-1}\frac{x}{a}+C(x>a>0)$$

Forms Involving $ax+b$

21. $$\int(ax+b)^ndx=\frac{(ax+b)^{n+1}}{a(n+1)}+C,n\neq -1$$

22. $$\int x(ax+b)^ndx=\frac{(ax+b)^{n+1}}{a^2}\bigg[\frac{ax+b}{n+2}-\frac{b}{n+1}\bigg]+C,n\neq -1,-2$$

23. $$\int(ax+b)^{-1}dx=\frac{1}{a}\ln|ax+b|+C$$

24. $$\int x(ax+b)^{-1}dx=\frac{x}{a}-\frac{b}{a^2}\ln|ax+b|+C$$

25. $$\int x(ax+b)^{-2}dx=\frac{1}{a^2}\bigg[\ln|ax+b|+\frac{b}{ax+b}\bigg]+C$$

26. $$\int\frac{dx}{x(ax+b)}=\frac{1}{b}\ln\bigg|\frac{x}{ax+b}\bigg|+C$$

27. $$\int(\sqrt{ax+b})^ndx=\frac{2}{a}\frac{(\sqrt{ax+b})^{n+2}}{n+2}+C,n\neq -2$$

28. $$\int\frac{\sqrt{ax+b}}{x}dx=2\sqrt{ax+b}+b\int\frac{dx}{x\sqrt{ax+b}}$$

29.a. $$\int\frac{dx}{x\sqrt{ax+b}}=\frac{1}{\sqrt{b}}\ln\bigg|\frac{\sqrt{ax+b}-\sqrt{b}}{\sqrt{ax+b}+\sqrt{b}}\bigg|+C$$

29.b. $$\int\frac{dx}{x\sqrt{ax-b}}=\frac{2}{\sqrt{b}}\tan^{-1}\sqrt{\frac{ax-b}{b}}+C$$

30. $$\int\frac{\sqrt{ax+b}}{x^2}dx=-\frac{\sqrt{ax+b}}{x}+\frac{a}{2}\int\frac{dx}{x\sqrt{ax+b}}+C$$

31. $$\int\frac{dx}{x^2\sqrt{ax+b}}=-\frac{\sqrt{ax+b}}{bx}-\frac{a}{2b}\int\frac{dx}{x\sqrt{ax+b}}+C$$

Forms Involving $a^2+x^2$

32. $$\int\frac{dx}{a^2+x^2}=\frac{1}{a}\tan^{-1}\frac{x}{a}+C$$

33. $$\int\frac{dx}{(a^2+x^2)^2}=\frac{x}{2a^2(a^2+x^2)}+\frac{1}{2a^3}\tan^{-1}\frac{x}{a}+C$$

34. $$\int\frac{dx}{\sqrt{a^2+x^2}}=\sinh^{-1}\frac{x}{a}+C=\ln(x+\sqrt{a^2+x^2})+C$$

35. $$\int\sqrt{a^2+x^2}dx=\frac{x}{2}\sqrt{a^2+x^2}+\frac{a^2}{2}\ln(x+\sqrt{a^2+x^2})+C$$

36. $$\int x^2\sqrt{a^2+x^2}dx=\frac{x}{8}(a^2+2x^2)\sqrt{a^2+x^2}-\frac{a^4}{8}\ln(x+\sqrt{a^2+x^2})+C$$

37. $$\int\frac{\sqrt{a^2+x^2}}{x}dx=\sqrt{a^2+x^2}-a\ln\bigg|\frac{a+\sqrt{a^2+x^2}}{x}\bigg|+C$$

38. $$\int\frac{\sqrt{a^2+x^2}}{x^2}dx=\ln(x+\sqrt{a^2+x^2})-\frac{\sqrt{a^2+x^2}}{x}+C$$

39. $$\int\frac{x^2}{\sqrt{a^2+x^2}}dx=-\frac{a^2}{2}\ln(x+\sqrt{a^2+x^2})+\frac{x\sqrt{a^2+x^2}}{2}+C$$

40. $$\int\frac{dx}{x\sqrt{a^2+x^2}}=-\frac{1}{a}\ln\bigg|\frac{a+\sqrt{a^2+x^2}}{x}\bigg|+C$$

41. $$\int\frac{dx}{x^2\sqrt{a^2+x^2}}=-\frac{\sqrt{a^2+x^2}}{a^2x}+C$$

Forms Involving $a^2-x^2$

42. $$\int\frac{dx}{a^2-x^2}=\frac{1}{2a}\ln\bigg|\frac{x+a}{x-a}\bigg|+C$$

43. $$\int\frac{dx}{(a^2-x^2)^2}=\frac{x}{2a^2(a^2-x^2)}+\frac{1}{4a^3}\ln\bigg|\frac{x+a}{x-a}\bigg|+C$$

44. $$\int\frac{dx}{\sqrt{a^2-x^2}}=\sin^{-1}\frac{x}{a}+C$$

45. $$\int\sqrt{a^2-x^2}dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\sin^{-1}\frac{x}{a}+C$$

46. $$\int x^2\sqrt{a^2-x^2}dx=\frac{a^4}{8}\sin^{-1}\frac{x}{a}-\frac{1}{8}x\sqrt{a^2-x^2}(a^2-2x^2)+C$$

47. $$\int\frac{\sqrt{a^2-x^2}}{x}dx=\sqrt{a^2-x^2}-a\ln\bigg|\frac{a+\sqrt{a^2-x^2}}{x}\bigg|+C$$

48. $$\int\frac{\sqrt{a^2-x^2}}{x^2}dx=-\sin^{-1}\frac{x}{a}-\frac{\sqrt{a^2-x^2}}{x}+C$$

49. $$\int\frac{x^2}{\sqrt{a^2-x^2}}dx=\frac{a^2}{2}\sin^{-1}\frac{x}{a}-\frac{1}{2}x\sqrt{a^2-x^2}+C$$

50. $$\int\frac{dx}{x\sqrt{a^2-x^2}}=-\frac{1}{a}\ln\bigg|\frac{a+\sqrt{a^2-x^2}}{x}\bigg|+C$$

51. $$\int\frac{dx}{x^2\sqrt{a^2-x^2}}=-\frac{\sqrt{a^2-x^2}}{a^2x}+C$$

Forms Involving $x^2-a^2$

52. $$\int\frac{dx}{\sqrt{x^2-a^2}}=\ln|x+\sqrt{x^2-a^2}|+C$$

53. $$\int\sqrt{x^2-a^2}dx=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\ln|x+\sqrt{x^2-a^2}|+C$$

54. $$\int(\sqrt{x^2-a^2})^ndx=\frac{x(\sqrt{x^2-a^2})^n}{n+1}-\frac{na^2}{n+1}\int(\sqrt{x^2-a^2})^{n-2}dx,n\neq -1$$

55. $$\int\frac{dx}{(\sqrt{x^2-a^2})^n}=\frac{x(\sqrt{x^2-a^2})^{2-n}}{(2-n)a^2}-\frac{n-3}{(n-2)a^2}\int\frac{dx}{(\sqrt{x^2-a^2})^{n-2}},n\neq 2$$

56. $$\int x(\sqrt{x^2-a^2})^ndx=\frac{(\sqrt{x^2-a^2})^{n+2}}{n+2}+C$$

57. $$\int x^2\sqrt{x^2-a^2}dx=\frac{x}{8}(2x^2-a^2)\sqrt{x^2-a^2}-\frac{a^4}{8}\ln|x+\sqrt{x^2-a^2}|+C$$

58. $$\int\frac{\sqrt{x^2-a^2}}{a}dx=\sqrt{x^2-a^2}-a\sec^{-1}\bigg|\frac{x}{a}\bigg|+C$$

59. $$\int\frac{\sqrt{x^2-a^2}}{x^2}dx=\ln|x+\sqrt{x^2-a^2}|-\frac{\sqrt{x^2-a^2}}{x}+C$$

60. $$\int\frac{x^2}{\sqrt{x^2-a^2}}dx=\frac{a^2}{2}\ln|x+\sqrt{x^2-a^2}|+\frac{x}{2}\sqrt{x^2-a^2}+C$$

61. $$\int\frac{dx}{x\sqrt{x^2-a^2}}=\frac{1}{a}\sec^{-1}\bigg|\frac{x}{a}\bigg|+C=\frac{1}{a}\cos^{-1}\bigg|\frac{a}{x}\bigg|+C$$

62. $$\int\frac{dx}{x^2\sqrt{x^2-a^2}}=\frac{\sqrt{x^2-a^2}}{a^2x}+C$$

Trigonometric Forms

63. $$\int\sin axdx=-\frac{1}{a}\cos ax+C$$

64. $$\int\cos axdx=\frac{1}{a}\sin ax+C$$

65. $$\int\sin^2 axdx=\frac{x}{2}-\frac{\sin 2ax}{4a}+C$$

66. $$\int\cos^2 axdx=\frac{x}{2}+\frac{\sin 2ax}{4a}+C$$

67. $$\int\sin^n axdx=-\frac{\sin^{n-1}ax\cos ax}{na}+\frac{n-1}{n}\int\sin^{n-2} axdx$$

68. $$\int\cos^n axdx=\frac{\cos^{n-1}ax\sin ax}{na}+\frac{n-1}{n}\int\cos^{n-2} axdx$$

69.a. $$\int\sin ax\cos bxdx=-\frac{\cos(a+b)x}{2(a+b)}-\frac{\cos(a-b)x}{2(a-b)}+C,a^2\neq b^2$$

69.b. $$\int\sin ax\sin bxdx=\frac{\sin(a-b)x}{2(a-b)}-\frac{\sin(a+b)x}{2(a+b)}+C,a^2\neq b^2$$

69.c. $$\int\cos ax\cos bxdx=\frac{\sin(a-b)x}{2(a-b)}+\frac{\sin(a+b)x}{2(a+b)}+C,a^2\neq b^2$$

70. $$\int\sin ax\cos axdx=-\frac{\cos 2ax}{4a}+C$$

71. $$\int\sin^n ax\cos axdx=\frac{\sin^{n+1}ax}{(n+1)a}+C,n\neq -1$$

72. $$\int\frac{\cos ax}{\sin ax}dx=\frac{1}{a}\ln|\sin ax|+C$$

73. $$\int\cos^n ax\sin axdx=-\frac{\cos^{n+1}ax}{(n+1)a}+C,n\neq -1$$

74. $$\int\frac{\sin ax}{\cos ax}dx=-\frac{1}{a}\ln|\cos ax|+C$$

75. $$\int\sin^n ax\cos^m axdx=-\frac{\sin^{n-1} ax\cos^{m+1} ax}{a(m+n)}+\frac{n-1}{m+n}\int\sin^{n-2} ax\cos^m axdx,n\neq -m$$

76. $$\int\sin^n ax\cos^m axdx=\frac{\sin^{n+1} ax\cos^{m11} ax}{a(m+n)}+\frac{m-1}{m+n}\int\sin^n ax\cos^{m-2} axdx,n\neq -m$$

77. $$\int\frac{dx}{b+c\sin ax}=\frac{-2}{a\sqrt{b^2-c^2}}\tan^{-1}\bigg[\sqrt{\frac{b-c}{b+c}}\tan\bigg(\frac{\pi}{4}-\frac{ax}{2}\bigg)\bigg]+C,b^2>c^2$$

78. $$\int\frac{dx}{b+c\sin ax}=\frac{-1}{a\sqrt{c^2-b^2}}\ln\bigg|\frac{c+b\sin ax+\sqrt{c^2-a^2}\cos ax}{b+c\sin ax}\bigg|+C,b^2<c^2$$

79. $$\int\frac{dx}{1+\sin ax}=-\frac{1}{a}\tan\bigg(\frac{\pi}{4}-\frac{ax}{2}\bigg)+C$$

80. $$\int\frac{dx}{1-\sin ax}=\frac{1}{a}\tan\bigg(\frac{\pi}{4}+\frac{ax}{2}\bigg)+C$$

81. $$\int\frac{dx}{b+c\cos ax}=\frac{2}{a\sqrt{b^2-c^2}}\tan^{-1}\bigg[\sqrt{\frac{b-c}{b+c}}\tan\bigg(\frac{ax}{2}\bigg)\bigg]+C,b^2>c^2$$

82. $$\int\frac{dx}{b+c\cos ax}=\frac{1}{a\sqrt{c^2-b^2}}\ln\bigg|\frac{c+b\cos ax+\sqrt{c^2-a^2}\sin ax}{b+c\cos ax}\bigg|+C,b^2<c^2$$

83. $$\int\frac{dx}{1+\cos ax}=\frac{1}{a}\tan\bigg(\frac{ax}{2}\bigg)+C$$

84. $$\int\frac{dx}{1-\cos ax}=-\frac{1}{a}\cot\bigg(\frac{ax}{2}\bigg)+C$$

85. $$\int x\sin axdx=\frac{1}{a^2}\sin ax-\frac{x}{a}\cos ax+C$$

86. $$\int x\cos axdx=\frac{1}{a^2}\cos ax+\frac{x}{a}\sin ax+C$$

87. $$\int x^n\sin axdx=-\frac{x^n}{a}\cos ax+\frac{n}{a}\int x^{n-1}\cos axdx$$

88. $$\int x^n\cos axdx=\frac{x^n}{a}\sin ax-\frac{n}{a}\int x^{n-1}\sin axdx$$

89. $$\int\tan ax dx=\frac{1}{a}\ln|\sec ax|+C$$

90. $$\int\cot axdx=\frac{1}{a}\ln|\sin ax|+C$$

91. $$\int\tan^2 axdx=\frac{1}{a}\tan ax-x+C$$

92. $$\int\cot^2 axdx=-\frac{1}{a}\cot ax-x+C$$

93. $$\int\tan^n axdx=\frac{\tan^{n-1}ax}{a(n-1)}-\int\tan^{n-2} axdx,n\neq 1$$

94. $$\int\cot^n axdx=-\frac{\cot^{n-1}ax}{a(n-1)}-\int\cot^{n-1} axdx,n\neq 1$$

95. $$\int\sec axdx=\frac{1}{a}\ln|\sec ax+\tan ax|+C$$

96. $$\int\csc axdx=-\frac{1}{a}\ln|\csc ax+\cot ax|+C$$

97. $$\int\sec^2 axdx=\frac{1}{a}\tan ax+C$$

98. $$\int\csc^2 axdx=-\frac{1}{a}\cot ax+C$$

99. $$\int\sec^n axdx=\frac{\sec^{n-2}ax\tan ax}{a(n-1)}+\frac{n-2}{n-1}\int\sec^{n-2}axdx,n\neq 1$$

100. $$\int\csc^n axdx=-\frac{\csc^{n-2}ax\cot ax}{a(n-1)}+\frac{n-2}{n-1}\int\csc^{n-2}axdx,n\neq 1$$

101. $$\int\sec^n ax\tan axdx=\frac{\sec^n ax}{na}+C$$

102. $$\int\csc^n ax\cot axdx=-\frac{\csc^n ax}{na}+C$$

Inverse Trigonometric Forms

103. $$\int\sin^{-1} axdx=x\sin^{-1} ax+\frac{1}{a}\sqrt{1-a^2x^2}+C$$

104. $$\int\cos^{-1} axdx=x\cos^{-1} ax-\frac{1}{a}\sqrt{1-a^2x^2}+C$$

105. $$\int\tan^{-1} axdx=x\tan^{-1} ax-\frac{1}{2a}\ln(1+a^2x^2)+C$$

106. $$\int x^n\sin^{-1} axdx=\frac{x^{n+1}}{n+1}\sin^{-1} ax-\frac{a}{n+1}\int\frac{x^{n+1}dx}{\sqrt{1-a^2x^2}},n\neq -1$$

107. $$\int x^n\cos^{-1} axdx=\frac{x^{n+1}}{n+1}\cos^{-1} ax-\frac{a}{n+1}\int\frac{x^{n+1}dx}{\sqrt{1-a^2x^2}},n\neq -1$$

108. $$\int x^n\tan^{-1} axdx=\frac{x^{n+1}}{n+1}\tan^{-1} ax-\frac{a}{n+1}\int\frac{x^{n+1}dx}{1+a^2x^2},n\neq -1$$

Exponential and Logarithmic Forms

109. $$\int e^{ax}dx=\frac{1}{a}e^{ax}+C$$

110. $$\int b^{ax}dx=\frac{1}{a}\frac{b^{ax}}{\ln b}+C$$

111. $$\int xe^{ax}dx=\frac{e^{ax}}{a^2}(ax-1)+C,b\neq 1$$

112. $$\int x^ne^{ax}dx=\frac{1}{a}x^ne^{ax}-\frac{n}{a}\int x^{n-1}e^{ax}dx$$

113. $$\int x^nb^{ax}dx=\frac{x^nb^{ax}}{a\ln b}\int x^{n-1}b^{ax},b>0,b\neq 1$$

114. $$\int e^{ax}\sin bxdx=\frac{e^{ax}}{a^2+b^2}(a\sin bx-b\cos bx)+C$$

115. $$\int e^{ax}\cos bxdx=\frac{e^{ax}}{a^2+b^2}(a\cos bx+b\sin bx)+C$$

116. $$\int\ln axdx=x\ln ax-x+C$$

117. $$\int x^n(\ln ax)^mdx=\frac{x^{n+1}(\ln ax)^m}{n+1}-\frac{m}{n+1}\int x^n(\ln ax)^{m-1}dx,n\neq -1$$

118. $$\int x^{-1}(\ln ax)^m=\frac{(\ln ax)^{m+1}}{m+1}+C,m\neq -1$$

119. $$\int\frac{dx}{x\ln ax}=\ln|\ln ax|+C$$

Forms Involving $\sqrt{2ax-x^2},a>0$

120. $$\int\frac{dx}{\sqrt{2ax-x^2}}=\sin^{-1}\bigg(\frac{x-a}{a}\bigg)+C$$

121. $$\int\sqrt{2ax-x^2}dx=\frac{x-a}{2}\sqrt{2ax-x^2}+\frac{a^2}{2}\sin^{-1}\bigg(\frac{x-a}{a}\bigg)+C$$

122. $$\int(\sqrt{2ax-x^2})^ndx=\frac{(x-a)(\sqrt{2ax-x^2})^n}{n+1}+\frac{na^2}{n+1}\int(\sqrt{2ax-x^2})^{n-2}dx$$

123. $$\int\frac{dx}{(\sqrt{2ax-x^2})^n}=\frac{(x-a)(\sqrt{2ax-x^2})^{2-n}}{(n-2)a^2}+\frac{n-3}{(n-1)a^2}\int\frac{dx}{(\sqrt{2ax-x^2})^{n-2}}$$

124. $$\int x\sqrt{2ax-x^2}dx=\frac{(x+a)(2x-3a)\sqrt{2ax-x^2}}{6}+\frac{a^3}{2}\sin^{-1}\bigg(\frac{x-a}{a}\bigg)+C$$

125. $$\int\frac{\sqrt{2ax-x^2}}{x}dx=\sqrt{2ax-x^2}+a\sin^{-1}\bigg(\frac{x-a}{a}\bigg)+C$$

126. $$\int\frac{\sqrt{2ax-x^2}}{x^2}dx=-2\sqrt{\frac{2a-x}{x}}-\sin^{-1}\bigg(\frac{x-a}{a}\bigg)+C$$

127. $$\int\frac{xdx}{\sqrt{2ax-x^2}}=a\sin^{-1}\bigg(\frac{x-a}{a}\bigg)-\sqrt{2ax-x^2}+C$$

128. $$\int\frac{dx}{x\sqrt{2ax-x^2}}=-\frac{1}{a}\sqrt{\frac{2a-x}{x}}+C$$

Hyperbolic Forms

129. $$\int\sinh ax=\frac{1}{a}\cosh ax+C$$

130. $$\int\cosh ax=\frac{1}{a}\sinh ax+C$$

131. $$\int\sinh^2 axdx=\frac{\sinh 2ax}{4a}-\frac{x}{2}+C$$

132. $$\int\cosh^2 axdx=\frac{\sinh 2ax}{4a}+\frac{x}{2}+C$$

133. $$\int\sinh^n axdx=\frac{\sinh^{n-1} ax\cosh ax}{na}-\frac{n-1}{n}\int\sinh^{n-2} axdx,n\neq 0$$

134. $$\int\cosh^n axdx=\frac{\cosh^{n-1} ax\sinh ax}{na}+\frac{n-1}{n}\int\cosh^{n-2} axdx,n\neq 0$$

135. $$\int x\sinh axdx=\frac{x}{a}\cosh ax-\frac{1}{a^2}\sinh ax+C$$

136. $$\int x\cosh axdx=\frac{x}{a}\sinh ax-\frac{1}{a^2}\cosh ax+C$$

137. $$\int x^n\sinh axdx=\frac{x^n}{a}\cosh ax-\frac{n}{a}\int x^{n-1}\cosh axdx$$

138. $$\int x^n\cosh axdx=\frac{x^n}{a}\sinh ax-\frac{n}{a}\int x^{n-1}\sinh axdx$$

139. $$\int\tanh axdx=\frac{1}{a}\ln(\cosh ax)+C$$

140. $$\int\coth axdx=\frac{1}{a}\ln|\sinh ax|+C$$

141. $$\int\tanh^2 axdx=x-\frac{1}{a}\tanh ax+C$$

142. $$\int\coth^2 axdx=x-\frac{1}{a}\coth ax+C$$

143. $$\int\tanh^n axdx=-\frac{\tanh^{n-1} ax}{(n-1)a}+\int\tanh^{n-2} axdx,n\neq 1$$

144. $$\int\coth^n axdx=-\frac{\coth^{n-1} ax}{(n-1)a}+\int\coth^{n-2} axdx,n\neq 1$$

145. $$\int\operatorname{sech} axdx=\frac{1}{a}\sinh^{-1}(\tanh ax)+C$$

146. $$\int\operatorname{csch} axdx=\frac{1}{a}\ln\bigg|\tanh\frac{ax}{2}\bigg|+C$$

147. $$\int\operatorname{sech}^2 axdx=\frac{1}{a}\tanh ax+C$$

148. $$\int\operatorname{csch}^2 axdx=-\frac{1}{a}\coth ax+C$$

149. $$\int\operatorname{sech}^n axdx=\frac{\operatorname{sech}^{n-2}ax\tanh ax}{(n-1)a}+\frac{n-2}{n-1}\int\operatorname{sech}^{n-2}axdx,n\neq 1$$

150. $$\int\operatorname{csch}^n axdx=-\frac{\operatorname{csch}^{n-2}ax\coth ax}{(n-1)a}-\frac{n-2}{n-1}\int\operatorname{csch}^{n-2}axdx,n\neq 1$$

151. $$\int\operatorname{sech}^nax\tanh axdx=-\frac{\operatorname{sech}^n ax}{na}+C,n\neq 0$$

152. $$\int\operatorname{csch}^nax\coth axdx=-\frac{\operatorname{csch}^n ax}{na}+C,n\neq 0$$

153. $$\int e^{ax}\sinh bxdx=\frac{e^{ax}}{2}\bigg[\frac{e^{bx}}{a+b}-\frac{e^{-bx}}{a-b}\bigg]+C,a^2\neq b^2$$

154. $$\int e^{ax}\cosh bxdx=\frac{e^{ax}}{2}\bigg[\frac{e^{bx}}{a+b}+\frac{e^{-bx}}{a-b}\bigg]+C,a^2\neq b^2$$

Some Definite Integrals

155. $$\int_0^\infty x^{n-1}e^{-x}dx=\Gamma(n)=(n-1)!, n>0$$

156. $$\int_0^\infty e^{-ax^2}dx=\frac{1}{2}\sqrt{\frac{\pi}{a}},a>0$$

157. $$\int_0^{\pi/2}\sin^n xdx=\int_0^{\pi/2}\cos^n xdx= $$\begin{cases}\frac{1\cdot 3\cdot 5\cdots(n-1)}{2\cdot 4\cdot 6\cdots n}\cdot\frac{\pi}{2}, &\text{$n$ is an even integer} \geq 2\\\frac{2\cdot 4\cdot 6\cdots (n-1)}{3\cdot 5\cdot 7\cdots n}, &\text{$n$ is an odd integer} \geq 3\end{cases}$$ $$